In this paper, we establish several results concerning the approximation of periodic functions by Fejér and de la Vallée Poussin means in Lebesgue spaces L₂πᵖ. The obtained estimates are expressed in terms of function for L₂ and the second-order modulus of continuity. The approximation of periodic functions by trigonometric polynomials plays a central role in Fourier analysis. Among the classical summation methods, Fejér sums and de la Vallée-Poussin sums provide powerful tools for improving the convergence behavior of Fourier series. In this work, we investigate the approximation of -periodic functions in spaces L₂by these two summation methods. Special attention is given to the relationship between the smoothness of the function, measured via the second-order modulus of continuity, and the rate of approximation. Our results contribute to a clearer understanding of how summability methods refine Fourier approximation and provide effective tools for both theoretical and applied analysis.
Mikhael Shahoud (2024) studied this question.